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Mirrors > Home > MPE Home > Th. List > elxp6 | Structured version Visualization version GIF version |
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7913. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
elxp6 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp4 7913 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) | |
2 | 1stval 7977 | . . . . 5 ⊢ (1st ‘𝐴) = ∪ dom {𝐴} | |
3 | 2ndval 7978 | . . . . 5 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴} | |
4 | 2, 3 | opeq12i 4879 | . . . 4 ⊢ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ |
5 | 4 | eqeq2i 2746 | . . 3 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ↔ 𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩) |
6 | 2 | eleq1i 2825 | . . . 4 ⊢ ((1st ‘𝐴) ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵) |
7 | 3 | eleq1i 2825 | . . . 4 ⊢ ((2nd ‘𝐴) ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶) |
8 | 6, 7 | anbi12i 628 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) |
9 | 5, 8 | anbi12i 628 | . 2 ⊢ ((𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) |
10 | 1, 9 | bitr4i 278 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4629 ⟨cop 4635 ∪ cuni 4909 × cxp 5675 dom cdm 5677 ran crn 5678 ‘cfv 6544 1st c1st 7973 2nd c2nd 7974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fv 6552 df-1st 7975 df-2nd 7976 |
This theorem is referenced by: elxp7 8010 eqopi 8011 1st2nd2 8014 eldju2ndl 9919 eldju2ndr 9920 r0weon 10007 qredeu 16595 qnumdencl 16675 setsstruct2 17107 tx1cn 23113 tx2cn 23114 txhaus 23151 psmetxrge0 23819 xppreima 31871 ofpreima2 31891 smatrcl 32776 1stmbfm 33259 2ndmbfm 33260 oddpwdcv 33354 prproropf1olem0 46170 |
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